Finitely generated and not finitely generated rings

TL;DR

This paper explores the structure of intermediate rings between polynomial rings, providing combinatorial constructions of rings that are not finitely generated over a base polynomial ring, highlighting differences in ring generation.

Contribution

It introduces simple combinatorial methods to construct intermediate rings that are not finitely generated over polynomial rings, advancing understanding of ring extension properties.

Findings

Constructed intermediate rings between polynomial rings that are not finitely generated.

Demonstrated combinatorial methods for analyzing ring extensions.

Highlighted differences in finite generation properties of rings.

Abstract

Let $R_1$ be a commutative ring, let $R_2$ be a finitely generated extension ring of $R_1$, and let $S$ be a ring that is intermediate between $R_1$ and $R_2$. For $R_1 = R[x]$ and $R_2 = R[x,y]$, this paper gives simple combinatorial constructions of intermediate rings $S$ that are not finitely generated over $R[x]$.